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Source. Theorem 4, printed p. 12 (PDF p. 13) of the 22 May 2001 author manuscript.

Statement

There is a finite covering of the integers with moduli $m_1,\ldots,m_r$ such that:

1. for every positive integer $m$, at most three indices $\ell\in\{1,\ldots,r\}$ satisfy $m_\ell=m$;
2. every $m_\ell$ is odd and greater than $1$;
3. every $m_\ell$ has at least two distinct prime factors.

Proof pointer. The construction and its verification occupy the rest of Section 4, through printed p. 19 (PDF p. 20); the paper reports (p. 19) that the covering it builds uses 6928899 congruences. They were not reconstructed or independently checked here.

The paper observes (p. 12) that Theorem 4 gives an odd covering if each odd modulus may carry up to three congruences, and that a covering as in Theorem 4 with "three" replaced by "two" in (i) would give an $f(x)\in\mathbb Z^+[x]$ with $f(x)x^n+2$ reducible for all $n\geq0$.

Bears on. The result is a repeated-modulus relaxation of Problem 7. It does not give a covering with distinct moduli.